The volume of a slightly curved submanifold in a convex region
Author:
B. V. Dekster
Journal:
Proc. Amer. Math. Soc. 68 (1978), 203208
MSC:
Primary 53C40
MathSciNet review:
0474147
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Abstract: Let T be a compact convex region in an ndimensional Riemannian space, be the minimum sectional curvature in T, and be the minimum normal curvature of the boundary of T. Denote by a vdimensional sphere, plane or hyperbolic plane of curvature . We assume that , k are such that on there exists a circumference of curvature k. Let be its radius. Now, let Q be a convex (in interior sense) mdimensional surface in T whose normal curvatures with respect to any normal are not greater than x satisfying . Denote by the length of a circular arc of curvature x in with the distance between its ends. We prove that the volume of Q does not exceed the volume of a ball in of radius . These volumes are equal when T is a ball in and Q is its mdimensional diameter.
 [1]
B.
V. Dekster, Estimates for the volume of a domain in a Riemannian
space, Mat. Sb. (N.S.) 88(130) (1971), 61–87
(Russian). MR
0301671 (46 #827)
 [2]
B.
V. Dekster, An inequality of isoperimetric type for a domain in a
Riemannian space, Mat. Sb. (N.S.) 90(132) (1973),
257–274, 326 (Russian). MR 0362159
(50 #14601)
 [3]
B.
V. Dekster, Estimates of the length of a curve, J.
Differential Geometry 12 (1977), no. 1,
101–117. MR 0470906
(57 #10650)
 [4]
D.
Gromoll, W.
Klingenberg, and W.
Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in
Mathematics, No. 55, SpringerVerlag, BerlinNew York, 1968 (German). MR 0229177
(37 #4751)
 [1]
 B. V. Dekster, Estimates of the volume of a region in a Riemannian space, Math. USSRSb. 17 (1972), 6187. MR 0301671 (46:827)
 [2]
 , An inequality of the isoperimetric type for a domain in a Riemannian space, Math. USSRSb. 19 (1973), 257274. MR 0362159 (50:14601)
 [3]
 , Estimates of the length of a curve, J. Differential Geometry 12 (1977), 99115. MR 0470906 (57:10650)
 [4]
 D. Gromoll. W. Klingenberg and W. Meyer, Riemannische Geometrie im Grossen, Lecture Notes in Math., vol. 55 SpringerVerlag, Berlin and New York, 1968. MR 0229177 (37:4751)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197804741476
PII:
S 00029939(1978)04741476
Article copyright:
© Copyright 1978
American Mathematical Society
